Combine the ginger, lemon peel and 3 cups cold water in a medium saucepan over high heat. Bring to a boil and cook until reduced by half, about 5 minutes. Add 1 cup of the sugar and continue boiling until the sugar completely dissolves, 1 to 2 minutes. Strain the syrup into a small pitcher, cover tightly and refrigerate until very cold, at least 2 hours.
Combine the raspberries and the remaining 1 tablespoon sugar in a medium bowl. Let stand at room temperature, about 30 minutes.
Spoon some of the berries into the bottoms of 6 to 8 glasses and mash them gently with a spoon. Pour over some ginger syrup - more or less depending on how sweet you want your drink. Pour about 1/2 cup club soda in each glass, and serve with a straw.